A

Step 1: Consider the initial potential energy between the two spherical shells. Since both shells have charge Q and are a distance d apart, the potential energy (U) between the two charges is given by the formula for the potential energy between two point charges:

U = \frac{k \cdot Q \cdot Q}{d} = \frac{k Q^2}{d},

where k is Coulomb's constant (approximately 8.99 × 10^9 N m²/C²).

Step 2: As the two shells move apart due to the repulsion caused by the like charges, their potential energy will convert into kinetic energy. When they are at a large distance apart, the potential energy approaches zero.

Step 3: The law of conservation of energy states that the total energy (potential + kinetic) is conserved in the system. Initially, the total energy is solely potential, and when they are far apart, all of it has become kinetic.

Therefore, the total kinetic energy (K.E.) when they are at a large distance apart is equal to the initial potential energy:

K.E. (total) = U = \frac{k Q^2}{d}.

Step 4: As both shells are identical and start from rest, they will have equal amounts of kinetic energy. Therefore, the kinetic energy of each sphere when they are far apart would be half of the total kinetic energy:

K.E. (each) = \frac{1}{2} K.E. (total) = \frac{1}{2} \cdot \frac{k Q^2}{d} = \frac{k Q^2}{2d}.

Hence, the kinetic energy of each sphere when they are at a large distance apart is \frac{k Q^2}{2d}.
Therefore, the correct answer is option A.